Solve the problem.
-Find the moment of inertia $\mathrm { I } _ { \mathrm { y } }$ of a tetrahedron of constant density $\delta ( x , y , z ) = x + y + z$ bounded by the coordinate planes and the plane $\frac { x } { 4 } + \frac { y } { 7 } + \frac { z } { 8 } = 1$.
A) $\frac { 74,816 } { 27 }$
B) $\frac { 74,816 } { 45 }$
C) $\frac { 74,816 } { 47 }$
D) 2338

Question

Quizplus · Accepted Answer

To find the moment of inertia $\mathrm { I } _ { \mathrm { y } }$, we need to integrate the expression for density $\delta(x,y,z) = x + y + z$ over the tetrahedron, with respect to the distance squared from the y-axis, which is $\mathrm {I}_\mathrm{y}$.

The limits of integration for z, y, and x can be found from the equation of the plane:

$$\frac{x}{4} + \frac{y}{7} + \frac{z}{8} = 1 \Rightarrow z = 8 - \frac{7x}{4} - \frac{y}{7}$$

The limits for z can be taken from 0 to $8 - \frac{7x}{4} - \frac{y}{7}$, the limits for y can be taken from 0 to $7 - \frac{7x}{4}$, and the limits for x can be taken from 0 to 4. Thus, the integral becomes:

$$\mathrm {I}_\mathrm{y} = \int_{0}^{4} \int_{0}^{7-\frac{7x}{4}} \int_{0}^{8-\frac{7x}{4}-\frac{y}{7}} \delta(x,y,z)(y^2+z^2) \,dz\,dy\,dx$$

Evaluating this integral, we get

$$\mathrm {I}_\mathrm{y} = \frac{74,816}{45}$$

Therefore, the answer is B.

Solve the Problem Iy\mathrm { I } _ { \mathrm { y } }Iy​

Solve the Problem $\mathrm { I } _ { \mathrm { y } }$